Given the impossibility to compute higher order predictions for GWs observables following a simple approach based on multipole expansion, as shown in Sec.REF , on this chapter we want now to develop a framework that allow us to describe the slow inspiral phase of a Binary Coalescing System in classical General Relativity, and the successive emission of Gravitational Waves, using an Effective Field Theory approach (see [1]}, [2]}, [3]}, [4]}, [5]} for modern reviews ). Effective Field Theory methods (see [6]}, [7]}, [8]} for introductions) were first introduced in particle physics for the study of heavy quark field theory (see [9]}). They were then adapted to gravitational physics by Damour and Farése in [10]}, and later systematized by Goldberger and Rothstein in [11]}. Within this approach the action for a binary system can be evaluated using Feynman diagrams, with the use of modern scattering amplitudes multi-loop techniques for high precision calculations, as first seen in [12]}. During the slow inspiral phase the velocity \(v\sim r_s/r \ll 1\)  is a small parameter, and so the dynamics can be treated perturbatively. Hence, computations can be organized in the Post-Newtonian perturbation scheme, where General Relativity corrections are organized in powers of velocity \(v\) , and of Newton constant \(G_N\) , such that at \(n\) PN order we consider corrections that scale as: \(G_N^{n-l}l^{2(l)}\)  with \(0\le l\le n-1\) . In the treatment of the EFT approach of a binary coalescing system we will adopt the "mostly plus" convention: \(\eta _{\mu \nu }=diag(-,+,+,+)\) .
